no. 7
μ-constant monodromy groups and marked singularities
[Groupe de monodromie μ-constant et singularités marquées]
Hertling, Claus1
1 Universität Mannheim Lehrstuhl für Mathematik VI Seminargebäude A 5, 6 68131 Mannheim (Germany)
Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2643-2680.

Nous considérons d’un point de vue global les familles μ-constantes de germes de fonctions holomorphes à singularités isolées. Tout d’abord, nous étudions un groupe de monodromie des familles contenant une singularité fixée. Ce groupe est constitué d’automorphismes du réseau de Milnor qui respectent non seulement la forme d’intersection, mais aussi la forme de Seifert et la monodromie. Nous conjecturons qu’il contient tous les automorphismes de ce type, modulo ±id. Ensuite, nous définissons les singularités marquées et construisons leurs espaces de modules globaux pour leurs classes d’équivalence à droite. La conjecture sur le groupe impliquerait que ces espaces de modules sont connexes. Nous discutons de la relation avec les problèmes de type Torelli et nous formulons une nouvelle conjecture de type Torelli global pour les singularités marquées. Toutes ces conjectures sont montrées pour les singularités simples et pour 22 des 28 singularités exceptionnelles.

μ-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo ±id. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these moduli spaces are connected. The relation with Torelli type problems is discussed and a new global Torelli type conjecture for marked singularities is formulated. All conjectures are proved for the simple and 22 of the 28 exceptional singularities.

DOI : 10.5802/aif.2789
Classification : 32S15, 32S40, 14D22, 58K70
Keywords: $\mu $-constant deformation, monodromy group, marked singularity, moduli space, Torelli type problem, symmetries of singularities
Mots-clés : déformations $\mu $-constantes, groupe de monodromie, singularité marquée, espace des modules, problème de type Torelli, symétries de singularités

Hertling, Claus 1

1 Universität Mannheim Lehrstuhl für Mathematik VI Seminargebäude A 5, 6 68131 Mannheim (Germany) 
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Hertling, Claus. $\mu $-constant monodromy groups and marked singularities. Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2643-2680. doi : 10.5802/aif.2789. https://aif.centre-mersenne.org/articles/10.5802/aif.2789/

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